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13h
answered Adjoint pair of functors and cogenerator elements
13h
revised Adjoint pair of functors and cogenerator elements
deleted 3 characters in body
1d
comment Errata in Prof. Rotman AIHA book about projectives in the chain complex category?
No, there is no need to add anything to my last assertion. There is a notion of projective object in any category, and coproducts of projective objects – whenever they exist – are automatically projective.
1d
answered Errata in Prof. Rotman AIHA book about projectives in the chain complex category?
1d
comment Errata in Prof. Rotman AIHA book about projectives in the chain complex category?
@TobiasKildetoft The embedding theorem does not guarantee the preservation of infinite direct sums. If it did, then every abelian category with infinite direct sums would satisfy AB4.
1d
awarded  group-theory
2d
comment Is the category of monoids cartesian closed? Why?
Did you read the links you were given at MO?
Feb
7
comment Does $\operatorname{Spec}$ preserve pushouts?
It does not in general, but it does in this case. Compute explicitly.
Feb
7
comment Correct definition of model category
I'm not aware of any results that genuinely require functorial factorisation – however, doing without is sometimes a lot harder. See also here and here.
Feb
6
comment Is there a reasonable Grothendieck topology on the category of modules over a ring?
Incidentally, in the case of vector spaces over a field, the regular topology coincides with the topology in which only the maximal sieves cover.
Feb
6
comment Does $\operatorname{Spec}$ preserve pushouts?
The same example works.
Feb
6
revised Does $\operatorname{Spec}$ preserve pushouts?
added 83 characters in body
Feb
6
comment Is there a reasonable Grothendieck topology on the category of modules over a ring?
There are at least two: one where only maximal sieves are covering, and another where every sieve is covering.
Feb
6
comment Does $\operatorname{Spec}$ preserve pushouts?
This answers the OP's question: the corresponding cospan in $\mathbf{CRing}$ has a pullback, and $\operatorname{Spec}$ does not send it to a pushout.
Feb
6
answered Does $\operatorname{Spec}$ preserve pushouts?
Feb
6
revised Is there a reasonable Grothendieck topology on the category of modules over a ring?
edited tags
Feb
6
comment Is there a reasonable Grothendieck topology on the category of modules over a ring?
Much as in point set topology, we have minimal and maximal Grothendieck topologies. So it all depends on what you mean by "reasonable".
Feb
6
comment motivation for the direct limit
The claim about algebraic closures needs to be made more precise. While it is true that $\overline{\mathbb{F}_p}$ is (isomorphic to) the direct limit of its finite subfields, that diagram cannot be constructed without first having an algebraic closure. The issue is, of course, the existence of automorphisms of field extensions – merely taking the category of finite fields of characteristic $p$ will not give you the right diagram.
Feb
5
comment Name for categories with a certain property on coproducts
Er, any category of modules is an example...
Feb
5
comment Name for categories with a certain property on coproducts
The category is pointed, so you get morphisms by sending all the other summands to zero.